1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

Stochastic flows and stochastic differential equations

Outline: This is the first part of a two-semester sequence course on mathematical finance. In the Fall semester, we will examine the fundamental principles of financial derivatives from both financial and mathematical perspectives, and demonstrate how the mathematical tools from stochastic calculus, differential equations and numerical methods join forces to form an essential part in modern finance. The emphasis of the course is a mathematical understanding of the intrinsic relationships among various financial instruments, which serves as a basis for investment decisions and trading strategies. The central theme of the Fall semester is the classic Black-Scholes-Merton model, and we plan to give a thorough treatment of the original model, with extensive discussions on the practical extensions in response to various disadvantages of the original model. One of the most intuitive and transparent approaches to illustrate that is also extensively used in practice is the binomial tree model. It contains most of the essential idea of the Black-Scholes-Merton methodology, and it can be naturally extended to build more general continuous-time models. Time permitting, we will include as much real life examples as possible to make this a rewarding experience for those who plan to pursue a career in this direction, as well as those who are just intrigued by the subject and its impact on our society.

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Introduction to Mathematical Finance

Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations

L p estimates for fully coupled FBSDEs with jumps

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

A time-inconsistent stochastic optimal control problem with a recursive cost functional is studied. Equilibrium strategy is introduced, which is time-consistent and locally approximately optimal. By means of multiperson hierarchical differential games associated with partitions of the time interval, a family of approximate equilibrium strategy is constructed, and by sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB) equation is derived through which the equilibrium value function can be identified and the equilibrium strategy can be obtained. Moreover, a well-posedness result of the equilibrium HJB equation is established under certain conditions, and a verification theorem is proved. Finally, an illustrative example is presented, and some comparisons of different definitions of equilibrium strategy are put in order.

Time-Inconsistent Recursive Stochastic Optimal Control Problems

In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton--Jacobi--Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85--138. We emphasize that when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence...

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Optimal Control Problems of Fully Coupled FBSDEs and Viscosity Solutions of Hamilton--Jacobi--Bellman Equations

In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We study two cases of diffusion coefficients $\sigma$ of FSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle (DPP), and are viscosity solutions to the associated generalized Hamilton-Jacobi-Bellman (HJB) equations. The associated generalized HJB equations are related with algebraic equations when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE and doesn't depend on the control. For this we adopt Peng's BSDE method, and so in particular, the notion of stochastic backward semigroup in [16]. We emphasize that the fact that $\sigma$ also depends on $Z$ makes the stochastic control much more complicate and has as consequence that the associated HJB equation is combined with an algebraic equation, which is inspired by Wu and Yu [19]. We use the continuation method combined with the fixed point theorem to prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove some new basic estimates for fully coupled FBSDEs under the monotonic assumptions. In particular, we prove under the Lipschitz and linear growth conditions that fully coupled FBSDEs have a unique solution on the small time interval, if the Lipschitz constant of $\sigma$\ with respect to $z$ is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.

Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations

A maximum principle for fully coupled forward–backward stochastic control systems with terminal state constraints

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. For SDGs, the upper and the lower value functions are defined by the controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in [6], we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Furthermore, for a special case (when $\sigma,\ h$ do not depend on $y,\ z,\ k$), under the Isaacs' condition, we get the existence of the value of the game.

Qingmeng Wei

Papers

Time-Inconsistent Recursive Stochastic Optimal Control Problems

Optimal Control Problems of Fully Coupled FBSDEs and Viscosity Solutions of Hamilton--Jacobi--Bellman Equations

Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations

A maximum principle for fully coupled forward–backward stochastic control systems with terminal state constraints

Stochastic differential games for fully coupled FBSDEs with jumps